scipy.special.

chebyc#

scipy.special.chebyc(n, monic=False)[source]#

Chebyshev polynomial of the first kind on \([-2, 2]\).

Defined as \(C_n(x) = 2T_n(x/2)\), where \(T_n\) is the nth Chebyshev polynomial of the first kind.

Parameters:

nint: Degree of the polynomial.
monicbool, optional: If True, scale the leading coefficient to be 1. Default is False.

Returns:

Corthopoly1d: Chebyshev polynomial of the first kind on \([-2, 2]\).

See also

chebyt: Chebyshev polynomial of the first kind.

Notes

The polynomials \(C_n(x)\) are orthogonal over \([-2, 2]\) with weight function \(1/\sqrt{1 - (x/2)^2}\).

References

[1]

Abramowitz and Stegun, “Handbook of Mathematical Functions” Section 22. National Bureau of Standards, 1972.

Examples

Evaluate the Chebyshev polynomial \(C_3\) at \(x = 1\):

>>> import numpy as np
>>> from scipy.special import chebyc, chebyt
>>> np.isclose(chebyc(3)(1), -2.0)
True

The polynomial \(C_n\) is a scaled Chebyshev polynomial of the first kind:

>>> x = np.linspace(-2, 2, 5)
>>> np.allclose(chebyc(3)(x), 2 * chebyt(3)(x/2))
True

Plot \(C_n\) for several values of \(n\):

>>> import matplotlib.pyplot as plt
>>> x = np.linspace(-2, 2, 400)
>>> fig, ax = plt.subplots()
>>> for n in range(4):
...     ax.plot(x, chebyc(n)(x), label=rf"$C_{n}$")
>>> ax.set_title(r"Chebyshev polynomials $C_n$")
>>> ax.set_xlabel("x")
>>> ax.legend(loc="best")
>>> plt.show()

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